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Abstract
Optical solitons, particle-like excitations ubiquitous in many fields, can bind to form soliton molecules with striking molecule-like interactions. However, the exotic soliton interactions in soliton molecules are still largely unexplored in dual-wavelength mode-locked fiber lasers. Here, we reveal the dynamics of dissipative soliton molecules with periodic solitons collision in a dual-wavelength ultrafast fiber laser. The soliton molecules with a central wavelength of 1532.8 nm and 1561 nm exhibit conspicuously different evolution characteristics attributed to the difference in gain spectral intensity and trapped potential. The long-wavelength soliton molecule swiftly recovers to the initial state after collision, while the short-wavelength soliton molecule has a remarkable variation in temporal separation and operation state. Moreover, the multiple intensive repulsion and attraction in soliton molecule with energy transfer between leading and trailing solitons, and the formation of triplet soliton molecule in short-wavelength with multiple switching have also been observed. The different oscillating solutions coexisting in dual-wavelength soliton molecules involving oscillating and sliding phase evolution confirm the multistability of the dissipative system. These findings shed new insights into the dynamics of soliton molecules and solitons collision in nonlinear systems.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Solitons, as localized waves maintain their shape during propagation, have attracted numerous research attention in optics, fluids, Bose-Einstein condensates, and field theory [1–4]. In particular, the dissipative soliton, which relies on the double balance of gain and loss as well as nonlinearity and dispersion, exhibits diversified soliton interactions [5]. The mode-locked fiber lasers, as an absolutely dissipative system, suggest a wealth of intriguing soliton dynamics, i.e., soliton buildup [6,7], soliton molecule [8,9], rogue wave [10,11], soliton explosions [12,13], and breathing soliton [14,15]. In particular, the soliton molecule, as an analogy with matter molecules, exhibits fascinating particle-like interactions, e.g., vibration, synthesis, and dissociation, which constitute the fundamental problems in soliton physics [16–19]. In dissipative systems, oscillation structures in soliton tails that originate from dissipative effect will dramatically alter the scenarios of solitons interactions [20]. In addition to stationary soliton molecules, there are also dynamic soliton molecules, among which the time interval or phase difference between individual solitons changes over time, and breathing soliton molecules that experience periodic spectra and temporal evolutions [15,19].
The most dynamic evolution of the soliton molecule has been studied in single-wavelength mode-locked lasers. However, the exotic soliton interactions in soliton molecules are still largely unexplored in the dual-wavelength mode-locked fiber laser. As we know, dual-wavelength solitons periodically collision with each other due to the difference in group velocities at distinct wavelengths [21]. Then, a fundamental question remains elusive: whether the dual-wavelength soliton molecules exhibit similar evolution characteristics during the collisions. Meanwhile, solitons collision could induce various intriguing phenomena in ultrafast fiber lasers, such as weak pulses formation induced by cross-polarization coupling, soliton explosion, and rogue waves [22–26]. Moreover, dual-wavelength mode-locked fiber lasers have potential applications in many fields, e.g., dual-comb spectroscopy, optical communication networks, and optical sensing [27,28]. The wealthy nonlinear dynamics in the collision of dual-wavelength solitons may pave the way for the application in all-optical logic and universal computation [29]. Soliton molecules in dual-wavelength fiber laser may increase the capacity of telecommunications by providing a new coding scheme. From a fundamental standpoint, it is crucial to explore the soliton molecule evolution characteristics under collision perturbation to understand the general dynamics of soliton molecules in complex nonlinear systems. Therefore, more efforts need to be made in investigating the dynamics of dissipative soliton molecules in dual-wavelength ultrafast fiber lasers.
Here, we reveal the dynamics of dual-wavelength soliton molecules in an ultrafast fiber laser with an accurate time-stretch dispersive Fourier transform (TS-DFT) detection technique. Specifically, we defined the dual-wavelength soliton of central wavelengths at 1532.8 nm and 1561 nm as blue and red soliton, respectively [Fig. 1(b)]. The dual-color soliton molecules possess conspicuously different evolution characteristics during the collision that is attributed to the difference in gain spectral intensity and trapping potential. Meanwhile, the multiple intensive repulsion and attraction in blue soliton molecule with energy transfer between leading and trailing solitons, and the formation of blue triplet soliton molecule with multiple switching induced by soliton collision have also been observed. The different oscillating solutions coexisting in dual-color soliton molecules involving oscillating and sliding phase evolution confirm the multistability of the dissipative system. These results shed new light on the dynamics of dissipative soliton molecule and solitons collision.
Fig. 1. (a) Schematic of the dual-wavelength mode-locked fiber laser cavity and real-time detection system. (b) Average spectra of dual-wavelength single soliton record by an OSA.
2. Experimental setup
The experimental setup for dual-color solitons generation was a passively mode-locked Er-doped fiber laser based on nonlinear polarization rotation (NPR) [Fig. 1(a)]. The fiber cavity included two in-line polarization controllers (PC), an optical integrated module (OIM) that integrated a polarization-dependent isolator, a wavelength division multiplexer (WDM), and a 10:90 OC. The 2.5 m lowly-doped erbium-doped fiber (EDF) served as the gain medium and 24.5 m SM-28e fiber operated in an anomalous dispersion regime with a net cavity dispersion of ∼ −0.475 ps2. Two PCs are placed before and after the SMF to manipulate the polarization of light waves. By well tuning the PCs and pump power, the gain spectrum could be adjusted with two gain peaks corresponding to distinct wavelengths, leading to the dual-color soliton mode-locking state.
The temporal information of dual-color solitons was directly detected by a 20 GHz photodiode (Agilent 83440C) and digitized by a 20 GHz real-time oscilloscope (Lecroy SDA 820Zi-B), while the optical spectrum was recorded by an optical spectrum analyzer (OSA, YOKOGAWA 230 AQ6370D) and the real-time oscilloscope using TS-DFT simultaneously. The dispersive Fourier transform (DFT) branch was composed of a spool of dispersion compensating fiber (DCF) with −577 ps/nm (−182 ps/nm) dispersion and detected by a 12 GHz photodiode (New Focus 1544-B). The temporal and spectral resolution was 50 ps and 0.17 nm (0.30 nm), respectively [30]. The optical spectrum measured by OSA [Fig. 1(b)] corresponding to dual-color single soliton state, and the soliton molecule could be formed with increased pump power and maneuver polarization setting. The roundtrip time of blue (1532.8 nm) and red solitons (1561 nm) were 132.889 ns and 132.899 ns, respectively, which led to the periodical collision per 1.64 ms inside the cavity.
3. Results and discussion
3.1 Experimental results
The red or blue dissipative soliton molecules are generated by manipulation polarization setting and pump power, and the different evolution characteristics of dual-color soliton molecules could be resolved during collision. The red soliton molecule with oscillating behavior is generated at the pump power of 30 mW and possesses higher gain spectra intensity compared to the blue solitons. The red solitons in Fig. 2(a) contain an oscillating soliton molecule (OSM) and single stationary soliton. Here, the DFT dispersion is −182 ps/nm, thus the spectrum evolution of the soliton molecule and stationary single soliton can be observed separately without obscure. The DFT spectra evolution [Fig. 2(a)] indicates that the two blue solitons continuous collision with the red OSM and single stationary soliton. The zoom-in plot [Fig. 2(b)] of the dashed rectangle in Fig. 2(a) suggests the obvious dispersive wave (DW) shedding in red OSM during collision. The collision point as indicated by the horizontal dashed line [Fig. 2(b)], the obvious spectral shift of dual-color solitons could be observed during collision. The conspicuous drifting of shedding DW in red OSM could be resolved by the dashed rectangle in Fig. 2(b), and is further confirmed by the several typical single-shot spectra in Fig. 2(d). The shedding DW gradually drifts toward the longer wavelength in red OSM after collision [black arrow in Fig. 2(d)], and finally annihilates with roundtrips increased. Notably, a set of spectral fringes emerged on the blue soliton after collision [Fig. 2(b)], which attributed to the interference between the optical soliton and dispersive wave triggered by the cross-phase modulation (XPM) [31]. The obvious oscillating behavior could be found in the field autocorrelation trace evolution of the red soliton molecule, consistent with the spectra evolution in Fig. 2(a). The red OSM has a temporal separation of ∼ 10 ps before collision, and experienced single soliton transient annihilation and splitting during collision [dashed rectangle in Fig. 2(c)]. Then the red soliton molecule recovered to the initial temporal separation and oscillating state under the soliton reshaping effect.
Fig. 2. Different evolution characteristics in red/blue soliton molecule during collision. (a)-(d) Collision of red soliton molecule and blue soliton pair. (a) The DFT spectra evolution of dual-color solitons. (b) Zoom-in plot of (a). (c) The field autocorrelation traces calculated via the Fourier transform of red OSM spectra in (a). (d) Single-shot spectrum corresponding to several typical roundtrips in (b). (e)-(h) Collision of single red soliton and blue soliton molecule. (e) The temporal intensity evolution of dual-color solitons collision. (f) The corresponding DFT spectral evolution. (g) The corresponding field autocorrelation trace evolution in (f). (h) Zoom-in plot of the dashed rectangle in (f).
The higher gain spectra intensity at the short wavelength was achieved by variation polarization that leads to the formation of blue soliton molecule at the fixed pump power of 30 mW. The temporal intensity and DFT spectra evolution of dual-color solitons collision involving blue soliton molecule are shown in Figs. 2(e) and 2(f). The corresponding field autocorrelation trace indicates the blue solitons from stationary soliton pair with temporal separation 50 ps switching to an oscillating state with pulses interval significantly decrease to 17 ps [Fig. 2(g)]. The obvious interference fringe could be resolved in the spectrum evolution [Fig. 2(h)] further confirm the significant decrease of pulses interval in blue solitons after collision. The collision point is indicated by the horizontal dashed line. Notably, a set of spectral fringes could be observed after the blue solitons collision [dash rectangle in Fig. 2(h)], attributed to the interference between the optical solitons and dispersive wave. Here, the blue soliton pair with a temporal separation of 50 ps could also be regarded as a soliton molecule with binding interaction. The interference fringes could not be found in the DFT spectrum before collision attributed to the limit DCF dispersion −182 ps/nm. Meanwhile, no soliton transient annihilation and splitting were observed during the blue soliton molecule collision.
Further, the triplet blue soliton molecule was formed by slightly ramping up the pump power to 36 mW [Figs. 3(a) and 3(b)] with a higher gain spectra intensity in short-wavelength. The red single stationary soliton swift recovered to the initial state after collision, while the blue doublet OSM with a temporal separation of 10.2 ps transit to a triplet soliton molecule with a complex evolution process during collision. The new blue soliton was generated after collision from background noise, as indicated by the increased side peaks in autocorrelation trace [Fig. 3(b)]. Meanwhile, the new generated blue single soliton possesses narrower spectrum bandwidth compared with the doublet OSM [dashed rectangle in Fig. 3(a)], then the spectrum of the single blue soliton gradually broadens and interference with the doublet OSM, finally forming a triplet soliton molecule with irregular vibrate behavior. Although the pulse temporal separation in the blue soliton molecule is less than the temporal resolution of 50 ps which impedes direct observation of the pulse distribution in the time domain, the temporal pulse distribution can be determined by analytic fit according to the autocorrelation traces. In the analytic process, each soliton within the molecule was a sech-shape pulse with a temporal width of 500 fs. Here, the pulses intensity was neglected in the calculation, and the pulse temporal separation was inferred from the autocorrelation traces [dashed rectangle in Fig. 3(b)]. The analytical fit temporal pulse distribution [Fig. 3(c)] indicates that the three solitons in blue soliton molecule independently evolve with irregular vibrating. The leading soliton as a reference pulse in analytical fit, the trailing soliton suggests a larger temporal vibration amplitude compared to the middle soliton during evolution. The multiple switching between soliton triplet molecule (equal pulse separations) and (2 + 1)-type soliton molecule with unequal temporal separations can be identified by the red-hot spots in the autocorrelation trace [dashed rectangle in Fig. 3(b)]. Comparison between the experimental [Fig. 3(b)] and analytical fit autocorrelation traces [Fig. 3(d)], the good agreement demonstrates that the analytical fit results are satisfactory.
Fig. 3. (a)-(d) Formation of blue triplet soliton molecule. (a) The DFT spectra evolution of dual-color solitons collision. (b) The corresponding field autocorrelation trace evolution. The analytical fit in (c) temporal pulse distribution and (d) field autocorrelation trace corresponding to the dashed rectangle in (b). (e)-(h) Collision dynamics of blue soliton molecule with strong repulsive and attractive interactions. (e) The temporal intensity evolution of dual-color solitons collision. (f) The corresponding DFT spectral evolution. (g) Zoom-in plot of blue soliton molecule spectral evolution in (f). (h) The corresponding field autocorrelation trace evolution in (g).
Continue to maintain the higher gain spectrum intensity in blue solitons, the number of dual-color solitons could be further increased by ramping up pump power to 42 mW [Fig. 3(e)]. The blue soliton pair contains five solitons with a large temporal separation before collision, and the red soliton pair contains a single oscillating soliton and stationary soliton [inset in Figs. 3(e) and 3(f)]. The red solitons swiftly recovers to its initial state after collision. In contrast, the temporal separation of blue solitons suggests a conspicuous variation after each collision [inset in Fig. 3(e) corresponding to the dashed rectangle]. During the first collision with the red solitons, the first and second blue solitons have noticeable temporal repulsion. However, the third and fourth blue solitons suggest a clear decrease in temporal separation to form a doublet soliton molecule. To observe the detailed evolution of the third and fourth blue soliton, the zoom-in DFT spectra and corresponding field autocorrelation trace evolution are shown in Figs. 3(g) and 3(h). The pulses separation of two blue solitons swiftly decreases after first collision from 55 ps to 9.3 ps, then begin to bind tightly to form a doublet soliton molecule with irregular vibrate. The temporal separation of the blue soliton molecule gradually increases to a larger value during subsequent evolution. The multiple dramatic changes of pulses interval between 2.8 ps and 53 ps in blue soliton molecule could be resolved corresponding to intense attractive and repulsive interaction [dashed rectangle in Fig. 3(h)]. Such interaction attributes to the multiple energy transfer between the two pulses in blue soliton molecule, leading to the significant intensity difference between the leading and trailing solitons. Again, no soliton transient annihilation and splitting were observed in blue soliton molecule. The temporal separation of blue soliton molecule fast increases after second collision at RT 14050, and the corresponding interference fringe swiftly disappears in the DFT spectrum [Fig. 3(g)]. Finally, the blue soliton molecule transits to a stable soliton pair with a constant temporal separation of 80 ps.
With a close gain spectra intensity in dual-color solitons by well-tuning polarization, the collision of dual-color soliton molecule was resolved at a pump power of 65 mW. The dual-color solitons both contain a soliton molecule and single soliton during evolution, as indicated by the temporal intensity and DFT spectra evolution [Figs. 4(a) and 4(b)]. The temporal separation of red OSM does not significantly change during collisions [Figs. 4(d1) and 4(d2)], which swiftly recovers to the previous state after each collision and keeps almost the same pulses interval of ∼10 ps. The zoom-in plot [Fig. 4(e2)] of the dashed rectangle in Figs. 4(d1) and 4(d2) confirm the oscillating behavior of the red soliton molecule with an oscillating period of 62 RTs. Meanwhile, the temporal separation in blue solitons displays a remarkable variation during collision [Figs. 4(c1) and 4(c2)]. The pulses interval of blue soliton pair fast decreases after first collision and begins to bind to form a soliton molecule. The formed blue OSM with a temporal separation of 8.5 ps and an oscillating period of 78 RTs. The zoom-in plot [Fig. 4(e1)] of the dashed rectangle in Figs. 4(c1) and 4(c2) further confirm the oscillating behavior of blue soliton molecule. Then the temporal separation in blue soliton molecule swiftly increases after second collision, corresponding to the soliton molecule transit to a stable soliton pair with a temporal separation of 65 ps [Fig. 4(c2)].
Fig. 4. Collision dynamics of dual-color soliton molecule. (a) The temporal intensity evolution of dual-color solitons. (b) Corresponding DFT spectra evolution. (c1) The DFT spectra evolution of blue soliton molecule. (c2) The corresponding field autocorrelation trace evolution. (d1) The DFT spectra evolution of red soliton molecule. (d2) The corresponding field autocorrelation trace evolution. (e1) Zoom-in plot of the dashed rectangle in (c1) and (c2). (e2) Zoom-in plot of the dashed rectangle in (d1) and (d2). Trajectories of the internal degrees of freedom in the interaction space in (f1) blue OSM and (f2) red OSM corresponding to the black dashed line in (c1) and (d1), respectively, namely the inter-pulse separation τ and relative phase φ, the roundtrip number is displayed in color scale.
Further, the relevant internal degree of freedom for dual-color OSM is constituted by the pulse temporal separation τ and relative phase φ [32,33]. For the internal dynamics in blue OSM, the internal motion was visualized in 3D interaction space [Fig. 4(f1) corresponding to the black dashed line in Fig. 4(c1)], the relative phase mainly oscillates between two fixed turning points. For the internal dynamics in red OSM, the internal motion is shown in Fig. 4(f2), corresponding to the black dashed line in Fig. 4(d1). The sliding phase evolution is involved in red OSM, governed by the persistent intensity difference between each constituent. Meanwhile, a larger quasi-periodical modulation around 950 RT can be identified in the trajectory in interaction space [Fig. 4(f2)]. The difference in the oscillating period [62(78) RTs of red(blue) OSM], distinct temporal separation [10(8.5) ps of red(blue) OSM], and relative phase evolution of dual-color OSM further confirm the different oscillating solution can coexist in a multi-pulses situation that each soliton molecule evolves periodically in different ways.
To reproduce the main observed features of spectra evolution in dual-color OSM, we fit the phase evolution over cavity roundtrips with the help of a simple formula, while keeping the pulse widths fixed. For the spectra evolution of blue OSM with an oscillating phase evolution, the relative phase is modeled with the following equations: φ(z) = φ0 + Aφsin(z). The analytic fit phase evolutions shown in Fig. 5(c) (blue line) agree well with the experiment result [green line corresponding to RT 4000 to 4500 in Fig. 4(e1)]. The parameters are set as φ0 = 0, oscillation amplitude Aφ = 0.08π, and z = 0.18 πn, pulses temporal separation 8.5 ps, n being the roundtrip number. Each soliton in OSM is chosen to have a sech-shape profile with a temporal width of 300 fs. The analytic fit spectral and autocorrelation trace [Figs. 5(a) and 5(b)] reproduce the oscillating characteristic well of blue OSM [Fig. 4(e1)]. The relative phase in red OSM is modeled by the simple equation φ(z) = φ0 + Aφsin(z) + z. The sliding phase evolution in Fig. 5(f) (blue line) agree well with the experiment retrieved values (green line), with φ0 = 0, Aφ = 0.07π, z = 0.03 πn, and pulse temporal separation 10 ps. Based on these simple formulas, the analytic fit spectral and autocorrelation traces (Fig. 5) reproduces the experiment results [Figs. 4(e1) and 4(e2)] convincingly, confirming the distinct oscillating behavior of dual-color soliton molecules.
Fig. 5. Analytical fits for the dynamics of dual-color oscillating soliton molecule. (a) The spectral evolution of blue soliton molecule. (b) The corresponding autocorrelation trace evolution in (a). (c) Evolution of the relative phase for oscillating phase dynamics. (d) The spectral evolution of red soliton molecule. (e) The corresponding autocorrelation trace evolution in (d). (f) Evolution of the relative phase for sliding phase dynamics (blue and green line corresponding to analytical fits and experiment retrieve values, respectively).
3.2 Numerical simulation
To give more insight into dual-color soliton molecules evolution under collision perturbation, the numerical simulation was executed in a lumped propagation model. The dual-color solitons propagation within the fiber sections is modeled with a modified nonlinear Schrödinger equation for the slowly varying pulse envelope:
The simulated dual-color solitons evolution with Es =9.5 pJ, q0 = 0.3, Psat = 20 W [Fig. 6(a)] contains a blue single stationary soliton and red OSM reaches a good agreement with the experiment [Fig. 2(b)]. Kelly sideband in the spectrum of dual-color solitons transiently annihilates during collision, the new Kelly sidebands emerged and further shifted back to their initial wavelength under the soliton self-reshaping effect. The collision point is indicated by the red dashed line in the temporal evolution of red OSM [Fig. 6(b) corresponding to the dashed rectangle in Fig. 6(a)]. The leading soliton in red OSM transiently annihilates and transfers energy to the trailing soliton, which corresponds to the disappearance of interference fringe in the optical spectrum during collision. Then the trailing soliton split into two solitons with temporal separation fast increases and finally recovers to the initial pulse temporal separation and oscillating behavior [Fig. 6(c)], which is consistent with the experiment observation [Fig. 2(c)].
Fig. 6. Simulation of dual-color solitons collision. (a)-(c) The collision dynamics of red soliton molecule and single blue soliton. (a) Dual-color solitons spectra evolution. Temporal intensity evolution in (b) and the field autocorrelation trace evolution in (c) for red soliton molecule. (d)-(f) The collision dynamics of single red soliton and blue soliton molecule. (d) Dual-color solitons spectra evolution. Temporal intensity evolution in (e) and the field autocorrelation trace evolution in (f) for blue soliton molecule.
With a different initial condition and pump power at Es =9.7 pJ, q0 = 0.3, Psat = 20 W, the simulated spectra evolution of dual-color solitons containing a blue soliton molecule and single red soliton is shown in Fig. 6(d). The physical processes are analogous to the experimental observations [Figs. 2(e)–2(h)]. The collision point occurred as indicated by the red dashed lines in Fig. 6(e). The detailed temporal evolution of blue soliton molecule [Figs. 6(e) and 6(f)] indicate that the pulse separation firstly fast decreases and then increases after collision, corresponding to the intensive attractive and repulsive interactions. The stationary blue soliton molecule transit to a stable oscillating state with a smaller temporal separation [Fig. 6(f)], consistent well with the experimental result [Fig. 2(g)].
In this mode-locked fiber laser, the red and blue soliton molecule could be formed under versatile gain spectra distribution by manipulating polarization and pump power. The gain spectrum intensity at each wavelength could be swapped, producing various soliton assemble forms, e.g., single soliton, soliton pair, or soliton molecule. Further, the formation of triplet blue soliton molecule with multiple switching indicates the inelastic collision characteristic of dual-color solitons. The distinct oscillating solutions coexist of dual-color soliton molecules in the same dissipative system attributed to the unique dual-peak gain spectra characteristic. Moreover, the conspicuously different evolution of dual-color soliton molecule during collision is closely related to the trapping potentials, attributed to the difference in dispersive wave intensity. The dispersive wave tail covers adjacent pulses and perturbs them through XPM. The perturbation results in the pulses interaction within soliton molecule either attractive or repulsive. The dominant dispersive wave intensity in red soliton is significantly larger compared to the blue soliton, corresponding to a stronger trapping potential [34,35]. Thus, the red soliton molecule fast recovers to its initial state after collision, while the blue soliton molecule exhibits a more dramatic evolution process accompanied by multiple attractive and repulsive interactions [Figs. 4(c2) and 4(d2)].
4. Conclusion
In conclusion, we reveal the dynamics of dissipative soliton molecules in a dual-wavelength ultrafast fiber laser. The dual-color soliton molecule exhibit conspicuously different evolution characteristics during collision attributed to the difference in gain spectral intensity and trapping potential. The red soliton molecule swiftly recovered to the initial state after collision, while the blue soliton molecule displays a remarkable variation in temporal separation and operation state. Meanwhile, the multiple intensive repulsion and attraction in the blue soliton molecule with energy transfer have also been observed. The formation of triplet blue soliton molecule with multiple switching reveals the inelastic collision characteristic of dual-color solitons. The different oscillating solutions coexisting in dual-color soliton molecules involving oscillating and sliding phase evolution confirm the multistability of the dissipative system. It is important to note that soliton interaction in dual-color solitons is critically dependent on the gain spectral profile and could vary in different laser cavities, such as ultrafast Ti: sapphire lasers, passive fiber oscillators, and microresonators [36–38]. These findings provide new perspectives on the dynamics of soliton molecules and soliton collision in dissipative systems and pave the way toward the physical understanding of complex nonlinear science.
Funding
Health@InnoHK program of the Innovation and Technology Commission of the Hong Kong SAR Government.; Research Grants Council of the Hong Kong Special Administrative Region of China (CityU T42-103/16-N, HKU 17200219, HKU 17205321, HKU 17209018, HKU C7074-21GF).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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