We investigate Airy-soliton interaction in a nonlinear fiber with Raman effect. We find that Airy solitons may fuse upon interaction at a position that can be controlled by a proper engineering of the Airy tail direction. This control allows us to generate Airy solitons with varying deceleration. At variance with the case of two solitons interaction, Raman-induced soliton self-frequency shift (SSFS) is strongly enhanced when the leading soliton is replaced with the accelerating Airy pulse and slightly suppressed for the decelerating one. These notable features are ascribed to the unique properties of asymmetrical Airy pulses with a switchable direction of the oscillatory tails. We show the way these processes are uncovered unambiguously by cross-correlation frequency resolved optical gating. We also investigate the impact of chirp imposed on the input pulse on the SSFS dynamics. Our results not only provide a new way to manipulate the SSFS, but may help to improve the control of soliton fusion events during supercontinuum generation, optical rogue waves and giant dispersive waves formation.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Nonlinear pulse propagation in optical fiber is inevitably perturbed by higher-order linear and nonlinear effects, which leads to surprising and ubiquitous nonlinear phenomena such as supercontinuum generation  and optical rogue waves . Among higher-order nonlinear processes, intrapulse Raman amplification results from a non-instantaneous nonlinearity and it is particularly relevant for soliton interaction processes. This retarded nonlinear response leads to a continuous redshift of the pulse since low-frequency photons experience Raman amplification as the optical pulse propagates through the fiber. This Raman-induced soliton self-frequency shift (SSFS), was first experimentally discovered  and theoretically analyzed  for soliton pulses propagating in the regime of anomalous dispersion in 1986. However, the theoretical analysis of SSFS has been extended to non-solitonic pulses . Soon after its discovery, SSFS has been studied extensively for different fibers and pulse shapes with various applications including analog-to-digital converter , tunable femtosecond laser sources  and broadband supercontinuum generation .
The key challenge for these applications is controlling, enhancing or inhibiting the SSFS. Several methods have been proposed to manipulate the SSFS. Approaches such as upshifted filters , bandwidth-limited amplification , cross-phase modulation , self-steepening  and negative dispersion slope  are known to suppress the SSFS, whereas it can be enhanced by short pulses , topographic , and photonic crystal fibers .
More recently, following the first paper in quantum-mechanics in 1979 , self-accelerating Airy wave-packets were introduced in nonlinear fiber optics . Airy wave-packets have remarkable features such as self-healing  and quasi dispersion-free propagation . By engineering the sign of cubic phase modulation, one can create an Airy pulse with the oscillatory tail in front or behind its dominant peak and, correspondingly, the Airy pulse exhibits self-deceleration or self-acceleration . Furthermore, nonlinear propagation of Airy pulses shed solitons  and the interaction between Airy pulses can generate solitons and solitons pair [21,22], a fact that has inspired the use of these asymmetric pulse to manipulate supercontinuum generation [23,24], self-focusing dynamics [25,26] and SSFS [27–29]. The interaction between solitons and self-decelerating Airy pulses displays interesting dynamics owing to the tunable wavefront of the Airy. So far, however, the role of the Raman effect was not taken into account, and notably, only weak Airy pulses with respect to soliton were considered .
Which is the effect of the Raman amplification on the Airy-soliton interaction? Which are the differences between Airy-soliton interaction and soliton-soliton interaction? In particular, since the Raman response is very asymmetric and sensitive to the tail direction of Airy pulse , one can expect to control the Airy-soliton interaction exploiting the tail direction. However - to the best of our knowledge - the impact of the Raman effect on the Airy-soliton interaction has not been considered before.
Here we show that, in the presence of the Raman effect, Airy-soliton collisions can be largely controlled up to observe the fusion between a soliton and the Airy beam. As a result of the merger, a single robust structure with variable energy and acceleration appears. This finding demonstrates that Airy soliton interactions are not only beneficial for engineering the SSFS, but may also provide other degrees of freedom to control extreme nonlinear phenomena such as supercontinuum generation , rogue waves , and giant dispersive waves .
2. Theoretical model
We consider the dimensionless nonlinear Schrodinger equation (NLSE) with the inclusion of an intrapulse Raman term :Eq. (1), the amplitude is defined such that and is the time in the moving reference frame with group velocity measured in units of pulse width. is the normalized distance along the fiber, with dispersion length and second-order dispersion coefficient. , where and are, respectively, the peak power of incident pulse and the nonlinear parameters of the optical fiber. represents the relative strength of the Raman amplification. For the purpose of focusing the attention on the Raman effect, we prefer to neglect high-order dispersion terms and self-steeping effects.
A solution of Eq. (1) with is the Airy pulse,Eq. (2) correspond to decelerating and accelerating Airy wave-packets, respectively. On the other hand, for , Eq. (1) supports the celebrated soliton solution:
In our simulations, we consider in-phase input Airy and soliton pulses:
3. Numerical results
It is well known that, in the case of Eq. (1) without the Raman effect, two in-phase fundamental solitons attract each other and collide periodically, whereas two out-phase fundamental solitons repel and their interval increase monotonically with the propagation distance . This fundamental soliton interaction induced by the Kerr nonlinearity is very sensitive to their relative phase and separation. The periodic collapse resulting from solitons interaction can be avoided by increasing the soliton separation . However, when the Raman effect is taken into account, this classical dynamics drastically changes. Even in the in-phase case, the two fundamental solitons never collide and separate after an initial stage in which they weakly attract each other . The asymmetric nature of the intrapulse Raman interaction makes solitons gain different phase shifts. The effect is even more prevailing when higher-order solitons and Airy solitons are involved in the interaction process. In our simulation, the initial temporal separation is set equal to 20. For the case of a leading Airy pulse with tails behind its peak, to avoid its initial overlap with the soliton pulse and keep enough side lobes in the tail-trailing Airy pulse, we fixed the displacement of the Heaviside function to .
In Fig. 1 we report the temporal and spectral evolutions of the interaction between the soliton and higher-order solitons or Airy pulses in the presence of the Raman effect. For the case of two high-order soliton interaction, labeled to as S-S, each high-order soliton experiences an initial compression stage and shoots off an intense fundamental soliton. The emergent solitons propagate independently of each other and no longer experience collisions. This behavior can be understood by noting that the large separation between the incident high-order solitons makes the collision length longer, weakening the attractive interaction force between them. After an initial compression, the Raman effect begins to play a relevant role during the soliton propagation dynamics. Intense solitons slow down due to the Raman effect, which manifests its action through bending trajectories in time and red-shifted spectra in frequency domain. However, the leading soliton travels at a slower average speed than the trailing one. As the propagation distance further increases, the fast red-shifted soliton eventually catches up the other one. In particular, as the interval between of the two red-shifted solitons decreases, their overlap passes from frequency to time domain, as can be clearly observed from the temporal and spectral evolution. At the beginning, the red-shifted parts of the spectrum move in the same direction and present high-visibility interference fringes. This fact indicates that the two red-shifted solitons are mutually coherent and their spectra overlap. However, at a certain propagation distance, this overlap becomes very weak. The Raman effect impose a stronger influence on the leading red-shifted soliton than on the trailing one. As a result, their central wavelengths becomes different. The wavelength separation reaches the maximum when they are synchronized in time, a point where the two red-shifted solitons merge to form a single one whose deceleration was suddenly enhanced due to its high intensity and short width. These dynamics can be clearly observed form the peak intensity variations reported in Fig. 2(a).
The soliton fusion event can be controlled by replacing the leading incident soliton with the Airy pulse. Figure 1 shows soliton interaction with decelerating and accelerating Airy pulses. The temporal and spectral evolutions of the Airy-soliton interaction are very similar to that of the soliton case, although the fusion points change and the final frequency shift is different. For the case of decelerating Airy-soliton interaction (labeled as S-DEAP), the sign of in Eq. (4) is positive, and a relatively long propagation distance is required for the appearance of the fusion event, whereas the frequency shift is slightly smaller than that for soliton interaction. On the other hand, in the case of an accelerating Airy-soliton interaction (labeled as S-ACAP), the sign of Eq. (2) is negative, and the fusion event will take place after a shorter propagation distance. The fusion length is 5.49, which is almost half of that for two soliton interaction (). This fact strongly indicates that the fusion event is very sensitive to the accelerating Airy pulse.
In order to verify that the sudden deceleration takes place exactly at the fusion point, we track the interval between the two red-shifted solitons, as shown in Fig. 3(a). In the initial stage, the interval between the two red-shifted solitons is relatively stable. With a further increase in the propagation distance , however, the effect of Raman scattering effect becomes stronger and enhances the interaction. Consequently, the interval decreases rapidly, a feature clearly pointed out in Fig. 3(a). For comparison, we calculate the frequency centroid, which is defined as , a quantity very sensible to the spectral location of abrupt changes. The observed frequency centroids also confirm the characteristic features reported in Fig. 3(a). In all the three considered cases, the location of the fusion point is found in good agreement with the corresponding spectral mutational site. Moreover, as shown in Fig. 3(b), the SSFS is enhanced and suppressed for the accelerating and decelerating Airy pulse, respectively. This demonstrates that the Raman induced frequency shift can be manipulated using the symmetric and/or asymmetric nature of the interaction pulse.
To get a deep understanding of the differences between fusion events induced by input pulses with different profiles, we track the evolution of spectrograms, thus providing the simultaneous visualization of temporal and spectral changes. The spectrograms are obtained from the cross-correlation frequency resolved optical gating (X-FROG) trace. Mathematically, the X-FROG spectrograms is described as:Equation (5) gives the frequency components and their corresponding temporal locations along the propagation.
The spectrograms are presented in Fig. 4 at five different propagation distances for the three different cases. The corresponding animated versions are plotted in Visualization 1, Visualization 2 and Visualization 3 respectively. They uncover a quite clear picture of the entire process, especially in the vicinity of the fusion point. The left column of Fig. 4 and Visualization 1 show time-frequency distributions during the propagation of two interacting solitons. We can clearly see that the two spectral overlapping solitons first experience a spectra broadening ruled by self-phase modulation and the corresponding temporal compression. Then, the emergent solitons synchronously experience a frequency red-shift and interact with the remaining part on their right side. Their peak intensity changes are similar in the first three propagation distance, which corresponds to the overlapping of their peak intensity curves. As the propagation distance increases, the leading red-shifted soliton will interact with the dispersive residues shed by the trailing soliton: its peak intensity undergoes sharp oscillations before the fusion event appears. This interaction makes the peak intensity of the leading soliton larger than that of trailing one. These different mutual interactions are directly reflected in the peak intensity curves as a function of propagation distance, see Fig. 2(a). Therefore, they will meet each other at longer propagation distances. Once the overlap in time is completed, the two red-shifted solitons will merge into a structure with broad spectrum and large peak intensity that exhibits an immediate frequency acceleration supported by the Raman amplification [37,38].
When the asymmetrical Airy pulse leads the symmetrical soliton pulse, as shown in the middle and right columns of Fig. 4 as well as Visualization 2 and Visualization 3, the leading red-shifted soliton is limited to the characteristic tail direction of the Airy pulse. In Figs. 4(a2) and 4(a3) we can clearly observe that the spectrogram of the decelerating Airy pulse is opposite to that of the accelerating Airy pulse. Under the combined action of anomalous-dispersion and Kerr nonlinearity, a soliton is formed out of the main lobe of Airy pulse, and pulled out of the remaining part by the Raman-induced deceleration. Due to the self-healing ability of Airy pulse, the Airy pulse residue continues to accelerate or decelerate. Consequently, the emergent soliton originated from the tail-trailing Airy pulse collides with its side lobes due to opposite propagation directions, gaining more energy [Fig. 2(c)]. This process mimics an optical Newton's cradles . Differently, successive collisions do not happen for the case of tail-leading Airy pulse and the peak intensity varies with the propagation distance as shown in Fig. 2(b). The subsequent transmission process is consistent with the one occurring in the soliton case. When compared with the fusion point spectra, shown in the fourth row of Fig. 4, the spectral width of the fused soliton for the case of the interacting Airy pulse with trailing tail is notably larger than in the other two cases.
Hereafter, a brief physical explanation of the observations is given. We clearly observe energy exchanges in two pulse interaction owing to the inter-pulse Raman scattering effect. As well known, the Raman response function is the main source of Raman scattering. In our simulation, we use a linear approximation model for the Raman response function, which in the frequency domain can be written as , where is a slope parameter related to the Raman gain coefficient. Therefore, the gain coefficient increases linearly with frequency. We calculated the frequency differences between the two red-shifted solitons at the fusion point from Figs. 4(d1)–4(d3). The corresponding values are , , and respectively. The frequency difference is larger for the case of an input accelerating Airy pulse. In fact, in this case the soliton with longer wavelength gains more energy through Raman amplification and exhibits the strongest deceleration.
We know that the number of side lobes of an Airy pulse increases when the truncated coefficient is decreased. If the step function is not introduced to avoid overlap between the soliton tail, the separation between the two incident pulses must be increased. For instance, an Airy pulse with a displacement of 20 requires a smallest truncated coefficient . The corresponding spectral evolution is presented in Fig. 5(a). By comparing Fig. 5(a) and Fig. 1(h), it is evident that, for larger truncated coefficient, the fusion event requires a longer propagation distance and the final frequency red-shifted is reduced. Through a series of numerical experiments we find a specific dependence of the minimum separation on the truncated coefficient, as shown in Fig. 5(b).
Having described the control of SSFS by use of Airy soliton interaction, we will now study the impact of an initial frequency chirped Airy pulse on the SSFS dynamic. The linear propagation of chirped Airy pulse is determined by the signs of second-order dispersion and chirped parameter . As the chirp is imposed on the incident Airy pulse, its spectrum is transformed into an Airy distribution, whose tail depends on the sign of chirp. This broad spectrum leads to a partial overlapping between the soliton and chirped Airy pulse spectra. Carrying out a series of simulations on chirped Airy and soliton interactions, we find the frequency centroids evolutions reported in Fig. 6. It can be observed from Fig. 6 that the distance of the fusion point decreases with an increasing chirp. However, there are some apparent differences between the chirped accelerating Airy-soliton interaction [Fig. 6(a)] and the chirped decelerating Airy-soliton interaction [Fig. 6(b)]. In the former case, as shown in Fig. 6(a), after the fusion point, the frequency centroids will go to the same value. As the propagation distance is further increased, this tendency is reversed. The effect does not happen for the case of an interacting chirped accelerating Airy, shown in Fig. 6(b). However, it is more sensitive to the change of chirp values. The results show that chirp parameter can influence the SSFS dynamic, and should be fixed carefully for the control of SSFS in different cases.
In summary, we have numerically investigated the propagation of Airy and soliton pulses copropagating in a nonlinear optical fiber with Raman amplification. Both the incident soliton and Airy pulse are initially well-separated in time domain. Two fundamental solitons are found to emerge from the Airy and soliton pulses. The leading red-shifted soliton will interact with the dispersive remaining part shed by the trailing input pulse. For a leading input Airy pulse with trailing tail, the red-shifted soliton first collides with its side lobes, gains energy and then interact with the dispersive background shed by the trailing input pulse propagation. On the other hand, no collisions between the side lobe and the red-shifted soliton occur for decelerating Airy pulses. This different interaction process determines the specific fusion moment of the two red-shifted solitons. When two red-shifted solitons overlap in time, the energy transfer from the tailing to the leading one reach its maximum. As a result, the leading red-shifted soliton becomes more intense and, exhibiting sudden deceleration, leads to a rapidly accelerating SSFS. In addition, we also study the effect of the chirp Airy pulse on the control of SSFS. We find that the chirp parameter can also be used to enhance or suppress the SSFS, but its value needs to be carefully selected. Therefore, we expect that the symmetric and asymmetric pulses interaction may be useful for SSFS manipulation, and broadband supercontinuum emission. Our findings also point out the potential use of ultrafast pulse shaping technology to control well-known nonlinear phenomena from collision-induced optical rogue waves  to giant dispersive waves .
Natural Science Foundation of China (NSFC) (61505116); Natural Science Foundation of Guangdong Province (2016A030313049); Natural Science Foundation of SZU (000053); John Templeton Foundation (58277); Educational Commission of Guangdong Province (2016KCXTD006); China Postdoctoral Science Foundation (2018M630978).
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