Crossover from two-frequency pulse compounds to escaping solitons

The nonlinear interaction of copropagating optical solitons enables a large variety of intriguing bound-states of light. We here investigate the interaction dynamics of two initially superimposed fundamental solitons at distinctly different frequencies. Both pulses are located in distinct domains of anomalous dispersion, separated by an interjacent domain of normal dispersion, so that group velocity matching can be achieved despite a vast frequency gap. We demonstrate the existence of two regions with different dynamical behavior. For small velocity mismatch we observe a domain in which a single heteronuclear pulse compound is formed, which is distinct from the usual concept of soliton molecules. The binding mechanism is realized by the mutual cross phase modulation of the interacting pulses. For large velocity mismatch both pulses escape their mutual binding and move away from each other. The crossover phase between these two cases exhibits two localized states with different velocity, consisting of a strong trapping pulse and weak trapped pulse. We detail a simplified theoretical approach which accurately estimates the parameter range in which compound states are formed. This trapping-to-escape transition allows to study the limits of pulse-bonding as a fundamental phenomenon in nonlinear optics, opening up new perspectives for the all-optical manipulation of light by light.

distinct domains of anomalous dispersion, separated by an interjacent domain of normal dispersion. A mutual cross-phase modulation induced attractive potential provides the binding mechanism that holds the constituent pulses together 29 . This transfers the concept of a soliton induced strong refractive index barrier for a normally dispersive wave 30 , to the interaction of pulses in distinct domains of anomalous dispersion. The former process is enabled by a general wave reflection mechanism originally reported in fluid dynamics 31 , in optics referred to as the push-broom effect 32 , optical event horizon 33,34 , or temporal reflection 35 , allowing for a strong and efficient all optical control of light pulses 36,37 . This mechanism has been shown to naturally appear in the supercontinuum generation process [38][39][40][41] . The previously studied formation of molecule-like two-frequency pulse compounds constitutes a paradigmatic example of extreme states of light, also offering intriguing insights to atom-like features of a soliton, including its ability to act as a localized trapping potential with a discrete level spectrum 29 . For a higherorder nonlinear Schrödinger equation with positive 2OD and negative 4OD, similar compound states where recently also observed, and, along with the sech 2 -shaped single soliton solutions of earlier studies 9,12 , identified as members of a large family of generalized dispersion Kerr solitons 42 . Objects of this type have recently been observed within a mode-locked laser cavity 43 . Dual-frequency pulses with similar pulse structure have previously also been studied experimentally in passively mode-locked fiber lasers 44 , and in a model for dual-channel simultaneous modelocking based on the Swift-Hohenberg equation 45 . Further, two-color soliton microcomb states where reported in theoretical studies of Kerr microresonators in terms of the Lugiato-Lefever equation (LLE) with two separate domains of anomalous dispersion 46 , and in the standard LLE with added negative quartic group-velocity dispersion 47 . Bound states of distinct solitons, i.e. composite solitons, with a very similar pulse structure where reported in a combined theoretical and experimental study of the Kerr multistability in the LLE 48 . The properties of these kind of objects, which are referred to by a variety of names such as dual-frequency pulses 44 , two-color soliton states 46 , two-frequency soliton molecules 29 , composite solitons 48 , and, polychromatic soliton molecules 43 , are largely unexplored. Subsequently we refer to these objects simply as pulse compounds.
Here, we study the interaction dynamics of two initially superimposed fundamental solitons at distinctly different center frequencies in terms of a propagation constant for which the group velocity dispersion (GVD) has downward parabolic symmetry. Such a profile allows to parametrically define pairs of center frequencies at which the local dispersion parameters have the same absolute values at any order. This reduces the complexity of the underlying model and allows to explore the influence of the nonlinear interaction on the model dynamics more directly. Specifically, we here investigate how an initial group-velocity (GV) mismatch affects the formation of two-frequency pulse compounds. While it was shown that such compound states can compensate sufficiently small GV mismatches through excitation of internal degrees of freedom 29 , reminiscent of molecular vibrations, this puts their robustness to the test and sheds more light on the binding mechanism that holds the subpulses together. In the limit of large GV mismatch we observe a crossover from the formation of two-frequency compound states to escaping solitons. We demonstrate that the crossover region exhibits pulse compounds consisting of a strong trapping pulse and a weak trapped pulse, GV matched despite a large center frequency mismatch. Building upon the interaction of a single soliton with a localized attractive potential in terms of a perturbed NSE, we derive a simplified theoretical approach that suggests an analogy to classical mechanics and allows to accurately estimate the parameter range wherein pulse compounds are formed.
Equation (1) is free from the slowly varying envelope approximation but can be reduced to the generalized nonlinear Schrödinger equation by introduction of a complex envelope for a suitable center frequency 49 . By assuming γ = const. , it can further be reduced to a standard NSE with higher orders of dispersion. For the propagation of an initial field in terms of Eq. (1) we use a pseudospectral scheme implementing z-propagation using a fourth-order Runge-Kutta method 52 . Initial conditions. As pointed out above, the GVD is symmetric about ω 0 = 2 rad/fs . Two frequencies are group-velocity (GV) matched to ω 0 . In terms of the angular frequency detuning they are located at � GVM1,GVM2 = ± √ −6β 2 /β 4 ≈ ±0.926 rad/fs , specifying group-velocity matched frequencies at (ω GVM1 , ω GVM2 ) ≈ (1.074, 2.926) rad/fs , see Fig. 1a. Both frequencies are located in distinct domains of anomalous dispersion realized by the considered propagation constant, see Fig. 1b. In general, group-velocity matched co-propagation of anomalously dispersive light pulses is possible in the frequency ranges highlighted in red in Fig. 1a. More specifically, for the considered propagation constant, a mode in range ω ∈ (0.931 rad/fs, ω ZDW1 ) is GV matched to a mode in ω ∈ (ω ZDW2 , 3.069 rad/fs).
The solitons injected at ω 1 and ω 2 are subject to higher orders of dispersion, which, in principle, causes their velocities to slightly deviate from their bare group-velocities v g (ω 1 ) and v g (ω 2 ) , respectively 5,53 . For a soliton with center frequency ω s and duration t s , this might be taken into account by considering a "corrected" soliton velocity 54 For the full range of simulation parameters considered in the presented study, the largest relative difference of these velocities was found to be |v g − v ′ g |/v g < 10 −4 . Subsequently we opted to use the usual group-velocity v g when referring to the velocity of the initial solitons.
Propagation dynamics of limiting cases. Our earlier study of the interaction dynamics of initially overlapping group-velocity matched fundamental solitons with a vast frequency gap 29 , suggests that in the limiting case of group-velocity matched initial solitons [ �ω = 0 rad/fs ], a heteronuclear two-frequency pulse compound will form. The evolution of a corresponding initial condition in the propagation range z = 0−25 mm is shown in Fig. 2a. The composite pulse generated by this initial condition, highlighted in the spectrogram in Fig. 2b, consists of two subpulses with roughly similar amplitudes but distinctly different center frequencies. From the spectral intensity |E ω | 2 and the spectrogram P S , the vast frequency gap between both subpulses is clearly evident. It generates resonant radiation upon propagation and leads to a kind of "radiating" compound state. In Fig. 2b, these resonances are signaled by trains of nodes that separate from the localized state. A thorough analysis of a pulse compound with a similar composition was detailed in reference 29 [see Fig. 2(f) of that reference]. The binding mechanism that leads to the formation of such a composite pulse is realized by the mutual cross-phase modulation between its interacting subpulses 29 . The resulting pulse compounds are quite robust: small initial group-velocity mismatches can be compensated by frequency shifts of the subpulse center frequencies. This enables intriguing internal dynamics, reminiscent of molecular vibrations, examined more closely in Fig. 4 below. In the limiting case of a large group-velocity mismatch of the initial solitons, i.e. for large absolute values of �ω , we expect that both pulses escape their mutual binding. This is demonstrated for �ω = −0.17 rad/fs in Fig. 2e,f. As evident from the time-domain propagation dynamics in Fig. 2e, two separate localized states with nonzero relative velocity can indeed be identified. They can be distinguished well in the spectrogram in Fig. 2f, indicating no notable trapping by either pulse.
A crossover from the formation of two-frequency soliton compounds to escaping solitons can be expected based on two arguments. First, consider the point of view of mutual trapping of each pulse by a cross-phase modulation induced attractive potential formed by the other pulse 29 . Then, a classical mechanics interpretation of the propagation scenario suggests the existence of an escape velocity, sufficient for a particle to escape its trapping potential. We explore this analogy in more detail below. Second, for offset frequencies �ω > 0.143 rad/fs , i.e. ω 1 < 0.931 rad/fs and ω 2 > 2.926 rad/fs , no mode can be group-velocity matched to either initial soliton, see Fig. 1a. Having demonstrated the propagation dynamics for two specific values of the frequency offset parameter �ω , a thorough investigation of the crossover between the above limiting-cases in terms of �ω is in order.
Crossover from mutual trapping to escape. To better characterize the crossover from mutual trapping to unhindered escape of the initial solitons, we track the velocities of the dominant localized pulses in each domain of anomalous dispersion. In Fig. 3b, the asymptotic velocities associated with the initial solitons at ω 1 and ω 2 are labeled v 1 and v 2 , respectively. In relation to the two limiting cases illustrated earlier, we find that at �ω = 0 rad/fs (cf. Fig. 2a) the velocities of the compounds subpulses match each other and are in good agreement with the group-velocities of the initial solitons. At �ω = −0.17 rad/fs (cf. Fig. 2e) we find that the dominant pulses in each region of anomalous dispersion are clearly distinct, again in agreement with the groupvelocities of the initial solitons. In between, a sudden crossover occurs at �ω Matching subpulse velocities in the range �ω (−) c < �ω < 0 result from an initial transient propagation regime during which the mutual interaction of the initially superimposed pulses causes both pulse center frequencies to shift, thereby also changing the pulse spectrum. In this parameter range we observe that the soliton with higher amplitude, i.e. the soliton initially at ω 1 , assumes a dominant role. While the effect on this pulse is small, the effect on the pulse initially at ω 2 is rather large. This is shown in Fig. 4, where we detail a simulation run at �ω = −0.05 rad/fs . An initial transient behavior in range z < 10 mm is well visible, see Fig. 4a,b. In the latter, the initial velocity mismatch of both pulses induces a vivid dynamics. This is demonstrated in Fig. 4e, where the internal dynamics of the composite pulse in terms of the separation and relative-velocity of its subpulses, reminiscent of molecular vibrations, is shown. For this example we find the asymptotic frequency shifts ω 1 = 1.124 rad/fs → ω ′ 1 ≈ 1.113 rad/fs (Fig. 4c) and ω 2 = 2.876 rad/fs → ω ′ 2 ≈ 2.949 rad/fs (Fig. 4d). The frequency up-shift ω 2 → ω ′ 2 is expected to result in a pulse velocity for which v g (ω ′ 2 ) > v g (ω 2 ) (cf. Fig. 1a). More precisely, we find the velocity shift v g (ω 2 ) = 0.076868 µm/fs → v g (ω ′ 2 ) = 0.07695 µm/fs in agreement with the data shown in Fig. 3b. As evident from Fig. 4a, radiation is emitted predominantly in the initial stage of the pulse compounds formation process.
We find that in the vicinity of �ω (−) c , the asymptotic state is characterized by two distinct pulse compounds. The z-evolution of a corresponding initial condition at �ω = −0.1 rad/fs is shown in Fig. 2c,d. Therein, the timedomain propagation dynamics (left panel of Fig. 2c) shows two localized pulses that separate from each other for increasing propagation distance. As evident from the spectrogram at z = 25 mm (Fig. 2d), the two localized pulses are actually pulse compounds (labeled C1 and C2 in Fig. 2d), each consisting of a strong trapping pulse and a weak trapped pulse. An analogous phenomenon, referred to as development of a "soliton shadow", "mixing", www.nature.com/scientificreports/ or "soliton-radiation trapping", exists for coupled NSEs describing soliton propagation in birefringent fibers 21,25 , and gas-filled hollow-core photonic crystal fibers 55 . One of the main differences to other works is that we here allow for group velocity matching across a vast frequency gap, which plays an important role in observing this effect. For this reason, other studies of initially superimposed solitons with center frequency mismatch did not observe such an effect 56,57 . Figure 5 shows a more comprehensive analysis of the individual pulse compounds. As evident from Fig. 5a, the time-domain intensity of both pulse compounds exhibit a fringe pattern signaling the superposition of subpulses with a significant center frequency mismatch. In Fig. 5b,c (Fig. 5d,e), the spectrum of the compound labeled C1 [C2] is put under scrutiny. In either case, both subpulses are group velocity matched and a phase-matching analysis for the strong trapping pulse indicates no generation of resonant radiation 6,7 , see Fig. 5b,d. This is different from the radiating molecule in Fig. 2a. . (a) Point particle motion in an attractive potential. The particle can escape the well if its kinetic energy T class kin exceeds the potential depth U 0 (see text for details). Parameter range in which the particle cannot escape the well is shaded gray. Secondary ordinate shows the trapping coefficient C tr computed in a simplified model for a soliton interacting with a localized attractive potential (see text for details). (b) Comparison of observed asymptotic velocities v 1 and v 2 of the dominant localized pulses in the distinct domains of anomalous dispersion and corresponding propagation constant based group-velocities v g . Light-green solid and dashed lines indicate the group velocities v g (ω ′ 1 ) and v g (ω ′ 2 ) , obtained for the shifted pulse center frequencies ω ′ 1 and ω ′ 2 , respectively (see text for details). (c) Logarithm of the overlap parameter q at z = 25 mm , quantifying the degree of mutual trapping (see text for details). Shaded area beyond �ω ≈ 0.143 rad/fs indicates region in which group-velocity matching cannot be achieved, cf. c , v 2 crosses over to a value that follows the trend of v g (ω 2 ) , but exhibits the systematic deviation v g (ω 2 ) − v 2 ≈ 0.00007 µm/fs . This systematic deviation is again a consequence of the perturbation imposed by the presence of a superimposed pulse in the initial condition. As pointed out earlier, the direct overlap of two solitons at z = 0 mm leads to an initial transient stage, during which their mutual interaction causes both pulse center frequencies to shift. Here, the effect on the pulse initially at ω 1 is again small and the effect on the pulse initially at ω 2 is rather large. Analyzing the simulation run at �ω = 0.12 rad/fs , we find the frequency shifts ω 1 = 0.954 rad/fs → ω ′ 1 ≈ 0.961 rad/fs and ω 2 = 3.046 rad/fs → ω ′ 2 ≈ 2.991 rad/fs . The frequency down-shift ω 2 → ω ′ 2 is expected to result in a pulse velocity for which v g (ω ′ 2 ) < v g (ω 2 ) (cf. Fig. 1a). As evident from Fig. 3b, the pulse velocities v g (ω ′ 1 ) and v g (ω ′ 2 ) obtained for the shifted center frequencies are in excellent agreement with the observed pulse velocities (see light-green solid and dashed lines in Fig. 3b). As pointed out above, beyond �ω = 0.143 rad/fs , group velocity matching is not possible (shaded region in Fig. 3). This is reflected by the overlap parameter q, dropping down to negligible values for �ω > 0.143 rad/fs . We observe a shift of both pulse center frequencies towards each other for �ω > 0 , while they shift away from each other for �ω < 0 (see the example detailed in Figs. 4c,d). This results in group-velocity matching in the domain where pulse compounds are formed, This is different from studies of the unperturbed NSE, where the center frequencies of initially overlapping solitons where reported to shift towards each other for any reasonable initial frequency separation 58 .
To clarify how the term ∝ γ 1 ω in the definition of γ (ω) [Eq. (3)] affects our observations, we repeated the above parameter study using the modified coefficient function γ (ω) = γ 0 . This setting can be reduced to a standard NSE with higher orders of dispersion (see "Methods" for details), similar to the model in which generalized dispersion Kerr solitons were studied recently 42 . Considering this simplified coefficient function, the above , where P 0 is the peak intensity of the strong trapping pulse and ω ′ C1 is its center frequency. Local extrema indicate group velocity matching with the strong trapping pulse. Frequencies at which resonant radiation might be expected are indicated by the roots of showing the initial spectrum at z = 0 mm (labeled A), full spectrum at z = 25 mm (labeled B), and spectra of the strong trapping pulse (labeled C) and weak trapped pulse (labeled D) of C1. (d,e) Same as (b,c) for C2. www.nature.com/scientificreports/ parameter study involves two initial solitons with matching dispersion lengths [ L D,1 = L D,2 ] and equal amplitudes [ A 1 = A 2 ]. As shown in Fig. 3e, across the region of compound state formation (i.e. for |�ω| < 0.065 rad/fs ), the asymptotic velocities v 1 and v 2 are not longer dominated by any particular pulse. Instead, the resulting composite pulse has velocity v 0 . This is, again, achieved by a shift of the pulses center frequencies during an initial transient stage. In comparison to the case where γ (ω) is modeled via Eq. (3), we find that the region of compound state formation is narrower. Despite the higher orders of dispersion featured by Eq. (1), the results reported in Fig. 3e are in good qualitative agreement with the interaction dynamics of initially overlapping, group-velocity mismatched solitons in a model of two nonlinearly coupled NSEs 56 . Also, a systematically smaller value of the overlap parameter q is evident in Fig. 3f. Let us comment on the characteristics of the pulse compounds in the vicinity of the crossover. The distinct features of C1 and C2 in Fig. 5 are solely due to the unsymmetry caused by the coefficient function γ (ω) given by Eq. (3). Considering the above modified coefficient function, we find that the spectra of C1 and C2 are simply related by symmetry, i.e. we can obtain C2 by inversion of C1 about ω 0 = 2 rad/fs.

Discussion
For the whole range of frequency offsets considered in our numerical simulations, we find that the observed velocity v 1 closely follows the group velocity v g (ω 1 ) . Both are associated with the initial fundamental soliton with the larger amplitude. We here find that the observed velocity v 2 can match v 1 in the range −0.075 rad/fs < �ω < 0.08 rad/fs , specifying the range within which heteronuclear pulse compounds are formed by the considered initial conditions. Outside this range, the formation of a single two-frequency soliton molecule is inhibited, with two localized pulses separating from each other and suppressed trapping for large absolute values of the frequency offset parameter. We found that we can estimate the domain of molecule formation in terms of a simplified theoretical approach (see "Methods" for details). In the latter, the dynamics of a two-pulse initial condition of the form of Eq. (5), governed by the nonlinear propagation equation Eq. (1), is approximated by the dynamics of a single pulse evolving under a nonlinear Schrödinger equation with localized attractive potential, given by Therein the complex envelope φ(z, τ ′ ) describes the dynamics of the subpulse with smaller amplitude, i.e. the subpulse at ω 2 . The potential well U(τ ′ ) is related to the subpulse with higher amplitude, i.e. the subpulse at ω 1 , and is given by U(τ ′ ) = −U 0 sech 2 (τ ′ /t 0 ) with potential depth , and τ ′ = t − β 1 (ω 1 )z . A similar approximation, for the special case of group-velocity matched propagation β ′ 1 = 0 , was recently used to demonstrate trapped states in a soliton-induced refractive index well 29 . Equation (7) suggests an analogy to a one-dimensional Schrödinger equation for a fictitious particle of mass m = −1/β ′ 2 , evolving in an attractive potential localized along the τ ′ axis. The relative velocity between the soliton and the potential is β ′ 1 . From a classical mechanics point of view we might expect that a particle, initially located at the potential center at τ ′ = 0 , escapes the potential well if its "classical" kinetic energy along the τ ′ -axis, given by exceeds the potential depth U 0 . In other words, for T class kin < U 0 we expect the particle to remain trapped by the potential. For the original model, defined by Eq. (1), this might be used to approximately estimate the domain in which compound states are formed. The results of this simplified theoretical approach are summarized in Fig. 3a,d, where T class kin and U 0 are shown as function of the frequency offset parameter �ω . For example, considering the setup with γ (ω) defined by Eq. (3), the condition T class kin < U 0 is satisfied for −0.068 rad/fs < �ω < 0.098 rad/fs (Fig. 3a). Despite the various simplifying assumptions that led to the above trapping condition, the estimated bounds for the domain of compound state formation are in excellent agreement with the observed bounds discussed above. In Fig. 3a,d we complement the findings based on the classical mechanics analogy by probing the trapping-to-escape transition of a soliton in a potential well in terms of Eq. (20) via numerical simulations. We therefore computed a trapping coefficient, defined by with N = |φ(0, τ ′ )| 2 dτ ′ for z = 10 mm . Both are in excellent qualitative agreement.
In conclusion, we showed that there exists a limit in the group-velocity mismatch of the constituents of a solitonic two-frequency pulse compound, above which its existence is not possible anymore. We clarified the breakup dynamics for the compound states beyond that limit, and showed that every constituent takes away parts of the radiation, again depending on the relative group velocities. The velocity of the pulse compound before the breakup is determined mostly by its "heaviest" component. More generally, our work demonstrates clearly the limits of stability of multicolor solitonic pulse compounds and we expect that the presented www.nature.com/scientificreports/ crossover-phenomenon will be useful for studying and understanding the break-up dynamics of more complex multi-frequency compounds, such as the recently demonstrated polychromatic soliton molecules 43 .
The time-domain representation of Eq. (20), given by constitutes the simplified model which allows to estimate the parameter range in which two-frequency pulse compounds are formed (see Discussion in the main text). Let us note that Eq. (21) represents a nonlinear Schrödinger equation with an attractive external potential of sech-squared shape. Similar model equations were previously used to study soliton-defect collisions in the nonlinear Schrödinger equation 59,60 , and interaction of matter-wave solitons with quantum wells in the one-dimensional Gross-Pitaevskii equation 61 . While these studies considered the collision of a soliton with an external attractive potential, our aim is here to understand the escape of a soliton from such a potential. (19) φ(z, τ ′ ) = ̟ φ ̟ e −i̟ τ ′ , with φ ̟ (z) = χ (2) ̟ (z)e −i β (2) 0 +β (1) 1 ̟ z , and τ ′ = τ − β (1) www.nature.com/scientificreports/ Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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